[Paper Review] Addendum to: Edge-Unfolding Nearly Flat Convex Caps

Joseph O’Rourke|arXiv (Cornell University)|Sep 2, 2017

Advanced Differential Equations and Dynamical Systems3 references3 citations

TL;DR

This addendum extends the result of O'Rourke (2017) by proving that a nearly flat acutely triangulated convex cap, when closed into a polyhedron by adding its convex polygonal base, can still be edge-unfolded into a non-overlapping simple polygon (a net). The proof relies on constructing an angle-monotone spanning forest and using a small-curvature approximation to ensure the base can be safely flipped out without overlap, even when no single edge of the cap's unfolding lies on the convex hull.

ABSTRACT

This addendum to [O'R17] establishes that a nearly flat acutely triangulated convex cap in the sense of that paper can be edge-unfolded even if closed to a polyhedron by adding the convex polygonal base under the cap.

Motivation & Objective

• To extend the edge-unfolding result for nearly flat acutely triangulated convex caps to include the case where the cap is closed into a polyhedron by adding its convex polygonal base.
• To resolve the challenge that the base cannot be attached to any edge of the cap's unfolding if all such edges lie inside the convex hull, which could cause overlap.
• To establish conditions under which the base can be safely flipped out from the unfolding without overlap, even when no hull edge is available for attachment.
• To formalize the geometric approximation of composite rotations along cut paths as a single rotation about a center-of-gravity point, ensuring minimal error under small curvature.

Proposed method

• Construct an angle-monotone spanning forest using four θ-monotone paths from an internal vertex q, where θ < 90°, to avoid overlap in the unfolding.
• Ensure the cone-gap g formed by the four θ-quadrants contains no internal vertices by selecting q as the vertex closest to the boundary ∂C.
• Use Lemma 1 to approximate the composite rotation of multiple small rotations along a cut path as a single rotation about a center-of-gravity point, with error δ bounded by (1/2)∑ℓiωi.
• Apply Lemma 2 to show that the approximate rotation center lies within the convex hull of the rotation centers, ensuring the base can be attached safely.
• Demonstrate that for sufficiently small curvature Ω, the error δ is small enough that the base can be flipped across any edge e that is locally and globally safe in the unfolding.
• Use a counterexample with a dodecagonal boundary and shallow-angle cut paths to show that no edge may lie on the convex hull, but the curvature approximation still allows safe attachment.

Experimental results

Research questions

• RQ1 Can a nearly flat acutely triangulated convex cap be edge-unfolded when its base is added to form a closed polyhedron?
• RQ2 Is there a geometric condition under which the base can be flipped out from the unfolding without causing overlap, even when no edge of the unfolding lies on the convex hull?
• RQ3 How small must the curvature of the cap be for the composite rotation error to be negligible, ensuring the base attachment is safe?
• RQ4 Can the composite effect of multiple small rotations along a cut path be approximated by a single rotation about a center-of-gravity point, and how accurate is this approximation?
• RQ5 Does the center-of-gravity approximation of composite rotations always lie within the convex hull of the rotation centers, ensuring geometric safety?

Key findings

• A nearly flat acutely triangulated convex cap with its convex base added can be edge-unfolded into a non-overlapping simple polygon, extending the result of O'Rourke (2017).
• The base can be safely flipped out from the unfolding even when no edge of the cap's unfolding lies on the convex hull, provided the curvature is sufficiently small.
• The error δ between the true composite rotation center and the center-of-gravity approximation is bounded by (1/2)∑ℓiωi, which approaches zero as the rotation angles ωi tend to zero.
• For caps with curvature Ω < πΦ² ≈ 0.28Δθ, the curvature is already small enough to satisfy the condition ω ≲ 2Δθ required for the approximation to hold.
• The center-of-gravity approximation of composite rotations lies within the convex hull of the rotation centers, ensuring that the base can be attached without overlap.
• A counterexample with a dodecagonal boundary and shallow-angle cut paths confirms that no hull edge may be available, but the curvature-based approximation still enables a valid unfolding.